Factorizing Simple Expressions
Given expressions that are being added or subtracted,
the goal of factorizing is to separate out the largest
possible common factor. This can be useful to simplify
fractional expressions, to better view the contents of
an expression, or to aid in solving equations.
Given an expression such as
the following are the general steps for its
factorization:
|
Step 1:
Reduce all numbers to their prime factorization.
|
|
Step 2:
Expand all exponential expressions.
| 2·3·a·a·a·b·b − 2·2·2·a·a·a·a·b·b·b·b |
|
|
Step 3:
Find numbers and
variables that are common to all expressions.
|
2·3·a·a·a·b·b − 2·2·2·a·a·a·a·b·b·b·b |
The largest possible set
of these is
called the Greatest common factor
(GCF)
|
Step 4:
Isolated the GCF in each expression.
| 2·a·a·a·b·b (3) −
2·a·a·a·b·b
(2·2·a·b·b) |
|
Step 5:
Use the distributive law to move the GCF outside
being sure to place parentheses around what remains.
| 2·a·a·a·b·b
(3 − 2·2·a·b·b) |
Note:
The expression is now
factorized however it is good form to write
the final factorization in exponential form.
|
|
Step 6:
Check the solution
| 2·a3
b2(3
− 22
·a2)
= 2·a3
b2(3)
− 2·a3
b2(22
·a2)
= 6a3
b2
− 23
a4
b4
= 6a3
b2
− 8a4
b4 |
|
Factorize
| 12a
b2
c3
− 8b3
c + 4a4
b
c2 |
Step 1:
Reduce all numbers to
their prime factorization.
| 2·2·3·a
b2
c3
− 2·2·2·b3c
+
2·2a4
b
c2
|
|
Step 2:
Expand all exponential
expressions.
| 2·2·3·a·b·b·c·c·c − 2·2·2·b·b·b·c
+ 2·2·a·a·a·a·b·c·c |
|
Step 3:
Find the GCF.
|
2·2·3·a·b·b·c·c·c −
2·2·2·b·b·b·c
+ 2·2·a·a·a·a·b·c·c, |
|
Step 4:
Isolate the GCF in each expression.
|
2·2·b·c·(3·a·b·c·c) −
2·2·b·c·(2·b·b)+
2·2·b·c·(a·a·a·a·c) |
|
Step 5: Use
the distributive law to move the GCF
outside being sure to place parentheses
around what remains.
|
2·2·b·c·(3·a·b·c·c −
2·b·b
+ a·a·a·a·c) |
Note:
The expression
is now factorized however it is good form
to write the final factorization in
exponential form.
|
Step 6: Check
the solution
| 4bc·(3abc2 −
2b2
+ a4
c)
= 4bc·(3abc2)
− 4bc·(2b2)
+4bc·(a4
c)
= 12ab2
c3
−
8b3c
+ 4a4bc2 |
|
Factorize
| 9x3
y2 z
− 3xyz + 6x2
y3
z |
Step 1:
Reduce all numbers to
their prime factorization.
|
3·3·x3
y2 z
− 3·xyz
+
2·3
x2
y3
z |
|
Step 2:
Expand all exponential
expressions.
| 3·3·x·x·x·y·y·z − 3·x·y·z
+ 2·3·x·x·y·y·y
·z |
|
Step 3:
Find numbers and
variables that are common to all
expressions.
| 3·3·x·x·x·y·y·z −
3·x·y·z+
2·3·x·x·y·y·y
·z |
then
the GCF is
|
Step 4:
Isolated the GCF in each expression.
| 3·x·y·z(3·x·x·y) −
3·x·y·z(1)+
3·x·y·z(2·x·y·y) |
|
Step 5: Use
the distributive law to move the GCF
outside being sure to place parentheses
around what remains.
| 3·x·y·z(3·x·x·y −
1+ 2·x·y·y) |
Note:
The expression
is now factorized however it is good form
to write the final factorization in
exponential form.
|
Step 6: Check
the solution
| 3xyz(3x2
y
+ 2x·y2
− 1)= 3xyz(3x2
y
) + 3xyz(2xy2)
− 3xyz(1)
= 9x3
y2 z + 6x2
y3
z
−
3xyz
=
9x3
y2 z
−
3xyz + 6x2
y3
z |
|
 |